continued application of that improvement. With this viewpoint we ask again—what is the effect of a female life saved at age 0 to 4 as against one saved at age 55 to 59?
The female deaths at "age 0 to 4 in the United States in 1966 were 43,000, and the deaths at age 55 to 59 were about the same number—that is the reason I chose those ages for the illustration. Now we ask the question in the form, "What is the effect of a fall of 1/43,000 in the death rate at age 0 to 4 as against the same fall in the death rate at age 55 to 59, supposing that the new rates will continue?"
The Stationary Model
Let us first see what will be the effect on the probability I(a)/I0 of living to age a. We first note that the probability of living, /(a)//0, is equal to the exponential
= exp -
where //(r) is the death rate in the small interval from t to t + dt, A rise in the death rate ju(f) over the age interval x to x + 1, x < a, equal to A/^., will add this quantity to the integral in the exponential, so that it becomes / H(f)dt + &tJ-x- Hence I(a)/I0 contains the further factor e~AM* for alia > x. Approximately, an increase in the death rate of given amount A/i^ over the single year of age x diminishes the probabiliyt of living to subsequent ages a in a ratio.
Thus the decline of 1/43,000 at age 0 to 4 increases the proportion living to all later ages in the ratio 1 + 5/43,000, and a decline of 1/43,000 at age 55 to 59 increases the proportion living to ages greater than 60 in the same ratio 1 + 5/43,000.
Where mortality is already low, so that I(a)jl0 is nearly unity to the end of reproduction, further declines in the younger age-specific death rates have little effect on reproduction. With the rates prevailing in the United States, a 100 percent drop in all mortality up to age 50, so that everyone born lives to that age, would increase long-run growth by about 4 percent. The situation was very different in the developing countries of 20 years ago, when the fall in mortality greatly increased the rate of growth and tended to make the population much younger.
The effect of a change Aju^. on e0 may be inferred as a further step. If the probability that a child just bom will survive to age x is /(x)//0, then thehousand; the crude death rate is in the short run entirely insensitive to the ages at which those deaths occur, whether before, during, or long after the time of reproduction, whether of males or of females.